论文标题
在容量和扭转僵化方面
On capacity and torsional rigidity
论文作者
论文摘要
我们研究了形状功能的极端属性，这些功能是牛顿容量$ \ cp（\ overline {\ om}）$的产物，以及扭力刚度$ t（\ om）$的功率，用于开放式套装$ \ om \ om \ om \ subset \ r^d $，并带有紧凑型关闭$ \ operline $ \ om \ om om cline {\ om} $ and progecuge and progecuge。 It is shown that if $\Om$ is convex then $\cp(\overline{\Om})T^q(\Om)$ is (i) bounded from above if and only if $q\ge 1$, and (ii) bounded from below and away from $0$ if and only if $q\le \frac{d-2}{2(d-1)}$.此外，如果$ q> 1 $或$ d = 3 $和$ q = 1 $，则存在产品的凸最大值。以$ q <\ frac {d-2} {2（d-1）} $的价格存在凸的最小值。如果$ q \ le 0 $，则该产品在所有有限的集合中都通过$ 1 $的球将其最小化。
We investigate extremality properties of shape functionals which are products of Newtonian capacity $\cp(\overline{\Om})$, and powers of the torsional rigidity $T(\Om)$, for an open set $\Om\subset \R^d$ with compact closure $\overline{\Om}$, and prescribed Lebesgue measure. It is shown that if $\Om$ is convex then $\cp(\overline{\Om})T^q(\Om)$ is (i) bounded from above if and only if $q\ge 1$, and (ii) bounded from below and away from $0$ if and only if $q\le \frac{d-2}{2(d-1)}$. Moreover a convex maximiser for the product exists if either $q>1$, or $d=3$ and $q=1$. A convex minimiser exists for $q< \frac{d-2}{2(d-1)}$. If $q\le 0$, then the product is minimised among all bounded sets by a ball of measure $1$.
